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Chapter 1: Problem 35

A student simplified the expression \(7 x-2(3-2 x)\) as shown. $$ \begin{aligned} 7 x &-2(3-2 x) \\ &=7 x-2(3)-2(2 x) \\ &=7 x-6-4 x \\ &=3 x-6 \end{aligned} $$ WHAT WENT WRONG? Find the cor- rect simplified answer.

Short Answer

Expert verified
The correct simplified answer is \(11x - 6\).

Step by step solution

01

- Apply the Distributive Property

First, apply the distributive property to the term \(-2(3-2x)\). This means multiplying \(-2\) with \(3\) and \(-2x\). Therefore, \(-2(3) = -6\) and \(-2(-2x) = 4x\).
02

- Rewrite the Expression

Replace the original terms inside the parenthesis with the results from applying the distributive property: \(7x - 6 + 4x\)
03

- Combine Like Terms

Combine the like terms \(7x\) and \(4x\) to get \(11x\). The expression now is \(11x - 6\). This completes the simplification.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
When you see a term being multiplied by an expression inside parentheses, you're dealing with the distributive property. The distributive property tells us to multiply the term outside the parentheses by each term inside the parentheses. For example, in the expression \( -2(3 - 2x) \), we need to distribute \( -2 \) to both \( 3 \) and \( -2x \). This is how it works: \[ -2(3) = -6 \] \[ -2(-2x) = 4x \] By doing this, we remove the parentheses and simplify the expression.
Combining Like Terms
After applying the distributive property, it鈥檚 crucial to combine like terms. Like terms are terms that contain the same variable raised to the same power. For example, \( 7x \) and \( 4x \) are like terms. We can add these together to simplify the expression further. In our example, after applying the distributive property and rewriting the expression, we have \( 7x - 6 + 4x \). Now combine the like terms: \[ 7x + 4x = 11x \] Now the expression is \( 11x - 6 \). By combining like terms, we make the expression simpler and easier to understand.
Step-by-Step Solutions
Breaking down a problem into smaller steps makes it easier to solve. Let鈥檚 revisit our original problem and go through the steps:

  • **Step 1 - Apply the Distributive Property:** We start by distributing \( -2 \) in \( -2(3 - 2x) \). This gives us \( -6 \) and \( 4x \).
  • **Step 2 - Rewrite the Expression:** Replace the original terms inside parentheses with the results. Now, we have \( 7x - 6 + 4x \).
  • **Step 3 - Combine Like Terms:** Finally, we combine like terms \( 7x \) and \( 4x \) to get \( 11x \). So, the expression simplifies to \( 11x - 6 \).
By following these clear steps, we ensure we simplify expressions correctly. And if something goes wrong, it's easier to identify where and correct it.

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Most popular questions from this chapter

The operation of division is used in divisibility tests. A divisibility test allows us to determine whether a given number is divisible (without remainder) by another number. An integer is divisible by 5 if its last digit is divisible by \(5,\) and not otherwise. Show that (a) \(3,774,595\) is divisible by 5 and (b) \(9,332,123\) is not divisible by 5

Solve each problem. Michigan is projected to lose 3 seats in the U.S. House of Representatives and Illinois 2 in \(2020 .\) The states projected to gain the most seats are California with 9 , Texas with \(5,\) Florida with \(3,\) Georgia with \(2,\) and \(\mathrm{Ari}-\) zona with \(2 .\) Write a signed number that represents the algebraic sum of these changes.

The two tables show the heights of some selected mountains and the depths of some selected trenches. Use the information given to answer. \( \)\begin{array}{|l|c|}\hline \text { Mountain } & {\text { Height (in feet) }} \\\ {\text { Foraker }} & {17,400} \\ {\text { Wilson }} & {14,246} \\\ {\text { Pikes Peak }} & {14,110} \\ \hline\end{array} $$ \begin{array}{|l|c|}\hline\text {} & {\text { Depth }( \text { in feet } } \\ {\text { Trench }} & {\text { as a negative number }} \\ {\text { Philippine }} & {-32,995} \\ {\text { Cayman }} & {-24,721} \\ {\text { Java }} & {-23,376} \\ \hline \end{array} $$ What is the difference between the height of Mt. Foraker and the depth of the Philippine Trench?

Perform each indicated operation. $$ -3+[(-5-8)-(-6+2)] $$

Decide whether each statement is true or false. \(-|-5| \geq-|-9|\)
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